The critical pulling force for self-avoiding walks

Self-avoiding walks are a simple and well-known model of long, flexible polymers in a good solvent. Polymers being pulled away from a surface by an external agent can be modelled with self-avoiding walks in a half-space, with a Boltzmann weight $y = e^f$ associated with the pulling force. This model is known to have a critical point at a certain value $y_c$ of this Boltzmann weight, which is the location of a transition between the so-called free and ballistic phases. The value $y_c=1$ has been conjectured by several authors using numerical estimates. We provide a relatively simple proof of this result, and show that further properties of the free energy of this system can be determined by re-interpreting existing results about the two-point function of self-avoiding walks.